Hidden Variables in Quantum Mechanics

If you do NOT measure/decohere a particle in some way, it exists ONLY as a probability wave--there is no "actual" number for each of its unknown quantity, just a probability of what it will be. There are no "hidden variables" that we don't know and can't find out--the particle has no "actual" value for position, just a probability distribution.

One of my friends insists that his quantum physics teacher told him that Quantum Mechanics in fact does have "hidden variables" and that all particles have definite numbers for everything--we just can't know them all at once with complete accuracy. This doesn't make any sense to me--I've read that particles aren't "actually" point particles with positions/velocities/spins that we can't know, but simply have a probability of existing in a particular state and aren't "actually" in a certain state before we measure it. Just like Schrödinger's Cat--who is both dead and alive at the same time until the box is opened (he isn't "actually" dead or alive and we just can't know).

So am I correct, and quantum mechanics has no "hidden variables", or is my friend right here?

You are right. Your friend, or his teacher, has made a common mistake in assuming that quantum uncertainty is just due to our inabilty to measure accurately or know precisely. But you have stood up for the correct view: Uncertainty (says QM) is a basic feature of reality, and the particle does not have a position or a momentum or a spin around some axis or whatever, between measurements. What is has, as again you have seen, is an amplitude to have any of these things, the amplitude will generate the probability to find the particle with such and such value of observalble. And amplitudes can be superposed.

You have stood up for the conventional view. Quantum mechanics does admit hidden variable interpretations, such as Bohmian mechanics, but they are not generally accepted. The reason is that any hidden variable theory has to have (at least) two undesirable features. Bell's theorem tells us that they must be nonlocal and the Kochen Spekker theorem tells us that they must be contextual.

However, is this really any worse than saying that a system doesn't have definite properties until it is measured? I am undecided on this point, but it is clear that most of the physics community has answered yes to this question.

I agree with Slyboy, "hidden variables" is a view, or interpretation of quantum physics. It has been niether proven nor disproven.

Not quite.

Before Bell's theorem, the argument on the existence (or non-existence of hidden variables) literally was based on a matter of taste. Bell's theorem, or in particular, Bell inequality, suggested a statistical measurment in which the test of such hidden variables can be made. This has been extended via the Clauser, Horne, Shimony, and Holt (CHSH) type system.

The caveat here is that there are various "specie" of what are known as "hidden variables". What Bell and CHSH test are the so-called LOCAL hidden variables (LHV). It would be misleading or even wrong to say that there have been no tests on the existence of such local hidden variables. A series of ever more accurate violation of CHSH inequalities from a number of experiments together indicate that there are ample evidence that there are no LHV.

Before Bell's theorem, the argument on the existence (or non-existence of hidden variables) literally was based on a matter of taste. Bell's theorem, or in particular, Bell inequality, suggested a statistical measurment in which the test of such hidden variables can be made. This has been extended via the Clauser, Horne, Shimony, and Holt (CHSH) type system.

The caveat here is that there are various "specie" of what are known as "hidden variables". What Bell and CHSH test are the so-called LOCAL hidden variables (LHV). It would be misleading or even wrong to say that there have been no tests on the existence of such local hidden variables. A series of ever more accurate violation of CHSH inequalities from a number of experiments together indicate that there are ample evidence that there are no LHV.

I am not a particle physicist.

First off, although there are behaviors that cannot be explained by 'nice' (more on this later) local hidden variables, there may well be behaviors that are readily explained by local hidden variables. So, while there may well be 'nice' local hidden variables, there must either be non-'nice' hidden variables, non-local hidden variables, or faster than light (i.e. non-local) interaction.

Actually, AFAIK the experimental results only indicate that the behavior cannot be explained by hidden local variables with measurable positive probability distributions. So, although the mainstream view is that there are no hidden variables, a more accurate description might be that the mainstream view is that models with local hidden variables are too complicated to be acceptable. There are hidden variable models, like many worlds, and IIRC Bohm's model which use non-local hidden variables, and a local hidden non-measurable model due to Pitowsky.

Since I'm not familiar with QM, I'm not sure, but it may be possible to eliminate non-local non-measurable models by experiment as well, but I don't think any experiments of that type have been made.

First off, although there are behaviors that cannot be explained by 'nice' (more on this later) local hidden variables, there may well be behaviors that are readily explained by local hidden variables. So, while there may well be 'nice' local hidden variables, there must either be non-'nice' hidden variables, non-local hidden variables, or faster than light (i.e. non-local) interaction.

Actually, AFAIK the experimental results only indicate that the behavior cannot be explained by hidden local variables with measurable positive probability distributions. So, although the mainstream view is that there are no hidden variables, a more accurate description might be that the mainstream view is that models with local hidden variables are too complicated to be acceptable. There are hidden variable models, like many worlds, and IIRC Bohm's model which use non-local hidden variables, and a local hidden non-measurable model due to Pitowsky.

Since I'm not familiar with QM, I'm not sure, but it may be possible to eliminate non-local non-measurable models by experiment as well, but I don't think any experiments of that type have been made.

I don't exactly understand what you just said.

First of all, this has NOTHING to do with "particle physics". Particle physicists do not normally perform experimental measurements such as this.

Secondly, I did stress that the LOCAL hidden variables are the ones being tested via the CHSH-type system. I made NO judgement regarding non-local hidden variables, much less interpret their existence/validity.

Thirdly, testing any "....non-measurable models by experiment..." sounds rather odd and oxymoronic to me.

Let's make it clear with regard to EPR type tests. An observation at one place should have no special correlation to a independent measurement at another place if the tested attributes have independent local reality. Perhaps they are both causally correlated to a third local variable? No, this can't be per the Aspect experients.

Although local realists like to think that the reason the results are perfectly correlated at certain angles is that you are "asking the same question" at both places (so of course you get the same answer)... all LHV theories also say that the correlations should have different values than they do in actual experimental tests at ALL OTHER angles.

If you look at the stats, you will clearly see that certain combinations of the outcomes are suppressed at various angles relative to the independent local reality predictions. For example, if the cos^2 45 degrees = .5 and the hypothesized "independent reality" (HV) orientation is not between this angle, then you should get a ++ at the detectors occasionally (since there is a >0 chance of such resolution at the detectors). This doesn't happen (do the math if you are not certain of this). Some cases are suppressed! Why? Because there is no observer-independent local reality.

So in actuality, a reading at one detector is ALWAYS related to the reading at the other. Either:

a) They now may both be causally connected to a third (the hypothetical hidden) variable, but that makes the theory non-local. That non-local variable could exist anywhere, even on the other side of the universe. That makes the possibility of a more complete specification of the system possible. QM is incomplete. Now all we have to do is locate such a variable to confirm.

Or:

b) they may be un-real in the classical sense. But a measurement at one spot still affects the combined statistics no matter what. That makes them observer dependent. So the "local" version is still dependent on the observer to cause the non-local collapse of the wave function. In this version no useful information is able to be transmitted faster than c. QM is as complete as it gets, but the observer is a fundamental part of the equation (in accordance with the predictions of the Heisenberg Uncertainty Relations).

An observation at one place affects the results at the other place, regardless of distance. The only stat that matters is the angle between the non-local observers (i.e. the spatially separated polarizers). All other considerations are irrelevant to the results.

Please let me know when the independent HV is found on the other side of the universe. I am not trying to be cynical - that is actually possible in some of the TOEs being offered up which hypothesize that some dimensions are "rolled up". In those theories, the distance from here to the opposite side of the universe is not so far if you take a shortcut through the rolled up dimension. In the meantime, the standard model (Copenhagen Interpretation) doesn't look too bad to me.

The "standard model", i.e. textbook quantum mechanics, is much closer to von-Neuman's approach to QM than it is to the Copenhagen Interpretation. I agree that it is a perfectly adequate approach for most practical purposes.

However, I think that if most physiscists actually read what Bohr, Heisenberg, Pauli et. al. wrote on the subject, then they would have a hard time agreeing with the Copenhahen interpretation.

Secondly, I did stress that the LOCAL hidden variables are the ones being tested via the CHSH-type system. I made NO judgement regarding non-local hidden variables, much less interpret their existence/validity.

What I was saying is that the experiments do not discount all 'local hidden variables' but that they indicate that 'local hidden variables' are insufficient to explain the experimental results.

Thirdly, testing any "....non-measurable models by experiment..." sounds rather odd and oxymoronic to me.

I meant non-measurable in the sense of 'non-measurable set' rather than in the sense of 'not quantifiable by experiment' the terminology is a bit unfortunate in this context. What I mean to say is that there are functions with the property that:
[tex]\int_a^bf(x)dx + \int_b^cf(x)dx \neq \int_a^cf(x)dx[/tex]

For certain [tex]a,b[/tex] and [tex]c[/tex]. This pathology is related to non-measurable sets, so I refer to the function as a non-measurable function.

Bell's theorem requires an integration over all possible states, and (effectively) assumes that
[tex]\int_a^bp(x)dx + \int_b^cp(x)dx = \int_a^cp(x)dx[/tex]
for some arbitrary non-negative probability function [tex]p[/tex]. Thus there is a tacit assumption that the probability function is 'measurable' so that the equality holds, and, it's possible to construct mathematical models of QM using these pathological state probability functions.

The "standard model", i.e. textbook quantum mechanics, is much closer to von-Neuman's approach to QM than it is to the Copenhagen Interpretation. I agree that it is a perfectly adequate approach for most practical purposes.

However, I think that if most physiscists actually read what Bohr, Heisenberg, Pauli et. al. wrote on the subject, then they would have a hard time agreeing with the Copenhahen interpretation.

Von Neumann formalized much of quantum measuremnt theory, butn from what I understand his own personal interpreation placed special importnace on the idea of a 'concious' observer.

What I was saying is that the experiments do not discount all 'local hidden variables' but that they indicate that 'local hidden variables' are insufficient to explain the experimental results.

Hum... but that is like saying the MM-experiment does not discount the ether, just that the ether is insufficient to explain the experimental result.

All LHV models are confined within CHSH's "|S| <= 2" limits.[1] If you dispute this, could you give me citations that have formulated this to the contrary.

I meant non-measurable in the sense of 'non-measurable set' rather than in the sense of 'not quantifiable by experiment' the terminology is a bit unfortunate in this context. What I mean to say is that there are functions with the property that:
[tex]\int_a^bf(x)dx + \int_b^cf(x)dx \neq \int_a^cf(x)dx[/tex]

For certain [tex]a,b[/tex] and [tex]c[/tex]. This pathology is related to non-measurable sets, so I refer to the function as a non-measurable function.

Bell's theorem requires an integration over all possible states, and (effectively) assumes that
[tex]\int_a^bp(x)dx + \int_b^cp(x)dx = \int_a^cp(x)dx[/tex]
for some arbitrary non-negative probability function [tex]p[/tex]. Thus there is a tacit assumption that the probability function is 'measurable' so that the equality holds, and, it's possible to construct mathematical models of QM using these pathological state probability functions.

But all one needs to do is actually measure the path from a to b and then b to c, and compare it with a to c. Path integral in non-conservative systems isn't something unusual. We teach students that in elementary E&M. However, the question here is whether the process of a to b and b to c are "measurable". If they aren't, then we are just speculating that there IS such a path and all arguments are going to simply be a matter of taste.

When there are already very strong violations of Bell (or more specifically CHSH) inequality,[2,3] it is rather difficult for to understand the idea that these results merely reflect the "inadequacy" of LHV.

Hum... but that is like saying the MM-experiment does not discount the ether, just that the ether is insufficient to explain the experimental result.

Actually, it's entirely possible that a more modern theory will involve an ether-like substance. We don't hear about luminous ether or phlogiston, but dark matter and dark energy are quite popular these days.

Similarly, an EPR experiment is not sufficient (by itself) to discredit the notion of hidden position or momentum variables. This may well be a problem of language - that you mean to refer to a specific type of LHV theory rather than all theories that involve local hidden variables.

All LHV models are confined within CHSH's "|S| <= 2" limits.[1] If you dispute this, could you give me citations that have formulated this to the contrary.

Resolution of the Einstein-Podolsky-Rosen and Bell Paradoxes, Physical Review Letters 48, 1299-1302 (1982).
On the web at http://edelstein.huji.ac.il/staff/pitowsky/Itamar%20Pitowsky_files/Paper%2001.pdf [Broken]
Also http://edelstein.huji.ac.il/staff/pitowsky/Itamar%20Pitowsky_files/Paper%2001.pdf [Broken]

Does not explicity address CHSH, but does provide an LHV model that Bell's theorem (and from what I can tell CHSH) does not apply to.

Actually, it's entirely possible that a more modern theory will involve an ether-like substance. We don't hear about luminous ether or phlogiston, but dark matter and dark energy are quite popular these days.

But here, the difference being that there WAS a definite definition for the classical ether, and that tests based on those properties have come up zilch. Just because you dress something new but give it the same name doesn't mean it's the same old thing. Dark matter and dark energy are certainly NOT the old classical ether theory. Not by a long shot.

Similarly, an EPR experiment is not sufficient (by itself) to discredit the notion of hidden position or momentum variables. This may well be a problem of language - that you mean to refer to a specific type of LHV theory rather than all theories that involve local hidden variables.

Resolution of the Einstein-Podolsky-Rosen and Bell Paradoxes, Physical Review Letters 48, 1299-1302 (1982).
On the web at http://edelstein.huji.ac.il/staff/pitowsky/Itamar%20Pitowsky_files/Paper%2001.pdf [Broken]
Also http://edelstein.huji.ac.il/staff/pitowsky/Itamar%20Pitowsky_files/Paper%2001.pdf [Broken]

Does not explicity address CHSH, but does provide an LHV model that Bell's theorem (and from what I can tell CHSH) does not apply to.

And you should also know that this publication isn't widely accepted either. Read the comments that followed this publication:

But here, the difference being that there WAS a definite definition for the classical ether, and that tests based on those properties have come up zilch. Just because you dress something new but give it the same name doesn't mean it's the same old thing. Dark matter and dark energy are certainly NOT the old classical ether theory. Not by a long shot.

As I said, I think this is primarly an issue of semantics. "Luminus ether" specifically refers to 'a conducting fluid for light' and a specific theory while "local hidden variable" is not necessarily specific to CHSH situations. So, while dark matter, and dark energy are certainly not phlogiston or luminous ether, they are all hidden variables - postulated unknown quantities introduced in order to make theories work.

So this view isn't widely shared nor accepted as of now.

Unfortunately I don't have ready access to those articles. I'll look them up the next time I'm at a university. I don't dispute that the theory is not mainstream, or that the theory is probably not useful, or even that such a theory would not be completely equivalent to other quantum mechanical theories and may well be unsuitable for QM use.

As I said, I think this is primarly an issue of semantics. "Luminus ether" specifically refers to 'a conducting fluid for light' and a specific theory while "local hidden variable" is not necessarily specific to CHSH situations. So, while dark matter, and dark energy are certainly not phlogiston or luminous ether, they are all hidden variables - postulated unknown quantities introduced in order to make theories work.

I disagree. Dark matter and dark energy are not "hidden variables". The fact that we postulate their existence AND consequences are already something different than the EPR-type hidden variables, be it local or not. Secondly, if they are truly hidden, then all the money already spent, and going to be spent, in the search for them would be wasteful. However, the theories that make use of their existence DO make predictions on their properties, and thus, testable. That's why we go on looking for WIMPs, etc., because their properties ARE defined, and we can go look to see if they are there or not. I certainly would not call them "hidden variables".

Actually, you should have used String Theory as something that is full of "hidden variables". It, and its variation, still can't produce anything that is testable, even in principle! As Steven Carlip of UC-Davis says "Nobody understands String Theory well enough to derive observational consequences". So there ya go! :)

Von Neumann formalized much of quantum measuremnt theory, butn from what I understand his own personal interpreation placed special importnace on the idea of a 'concious' observer.

Possibly so. What I am objecting to is the equivalence "Copenhagen Interpretation" = "Textbook quantum mechanics", which a lot of people seem to make. Many textbooks say that they are giving the Copenhagen Interpretation and then go on to give something that is much closer to the treatment in von-Neuman's book.

This leads many people to say that they think the Copenhagen Interpretation is reasonable. What they mean is that a thoroughly operational interpretation of quantum mechanics, remaining ambivalent or even denying the reality of everything referred to by the theory except measurement results, is reasonable. This I can agree with, but it is not the Copenhagen Interpretation.

Possibly so. What I am objecting to is the equivalence "Copenhagen Interpretation" = "Textbook quantum mechanics", which a lot of people seem to make. Many textbooks say that they are giving the Copenhagen Interpretation and then go on to give something that is much closer to the treatment in von-Neuman's book.

This leads many people to say that they think the Copenhagen Interpretation is reasonable. What they mean is that a thoroughly operational interpretation of quantum mechanics, remaining ambivalent or even denying the reality of everything referred to by the theory except measurement results, is reasonable. This I can agree with, but it is not the Copenhagen Interpretation.

I have to admit that is exactly what my textbook does, but it does allude to Von Neumann's role and his different interpreation.

Ou of interest can you detail the fundamentals of the Copenhagen Interpreation as originally stated.

Characterizing the Copenhagen interpretation is quite difficult, due to the subtle differences in the opinions of its originators and the fact that some of them wrote things that seem contradictory. In particular, Bohr's writings on the subject are notoriously difficult to understand. Anyway, here is what I see as the main points relevant for the distinction between Copenhagen and textbook.

1. Atomic and subatomic systems do not obey the laws of classical physics. Since they are not in the realm of our everyday experience, we had no reason to expect that they would.

2. Any experimental arrangement is necessarily described in terms of classical physics because that is in our realm of everyday experience, e.g. positions of detectors and the times at which they click. Therefore, any interpretation of quantum physics necessarily involves classical concepts, at least at this level.

3. The split between the quantum and classical part of the world (Heisenberg cut) is entirely arbitrary, i.e. it can be placed wherever is most convenient for explaining the experiment in question. However, it must be placed somewhere and the quantum description is meaningless without this.

4. The above, along with the mathematical formalism, leads to Bohr's notion of complementarity.

5. Bohr and Heisenberg never explicitly introduced the collapse postulate and their interpretation of the wavefunction (i.e. whether it is real or represents our knowledge) is somewhat ambiguous. It is fairly clear that they didn't think that there was a "measurement problem", since that only arises if you think there isn't a fundamental division of the world into "quantum" and "classical" parts. Modern treatments compare Bohr's philosophy with that of Kant, rather than the logical positivists.

The other main points of the Copenhagen interpretation are more or less identical to von-Neuman's treatment, e.g. interpretation of the uncertainty relation etc. However, von-Neuman's treatment is much more operational, avoiding statements about the reality of anything except measurement results. In this way, it is much more positivist than Copenhagen. He also introduced the measurement problem.

Incidentally, Spekkens and Sipe are currently writing a textbook on quantum foundations, which will deal with all the major interpretations and will characterize the difference between Copenhagen and textbook much more carefully than I have been able to.